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jfy4
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I'm really excited to get this as a homework problem. I have wanted to feel good about this formalism is quantum mechanics for a while now but my own stupidity has been getting in the way... With this homework problem hopefully I can move on to a new level.
The most general observable is a density matrix. Generally it is a non-negative self-adjoint operator with trace 1. It has the general form
<br /> \rho=\sum_{n}p_n |n\rangle\langle n|<br />
where p_n is a classical probability distribution (\sum_{n} p_n=1,\; 0\leq p_n \leq 1) and |n\rangle\langle n| are projection operators that are not necessarily orthogonal. \rho represents a classical statistical ensemble of quantum states where the state |n\rangle appears with probability p_n. The ensemble average of an operator O is an ensemble of states described by a density matrix \rho is
<br /> \langle O \rangle_{\rho}=\mathbf{Tr}(O\rho )<br />
Physically this is the average of a number of measurements of O in a classical probability distribution of different states. Consider a polarized beam of protons where 30% of the protons have spin up in the x-direction and 70% have spin down in the z direction. Find the density matrix for this ensemble and compute the ensemble average of s_z in this ensemble of protons.
<br /> \mathbb{I}=\sum_{n}|n\rangle\langle n|<br />
I set up the density matrix like this
<br /> \rho=\frac{3}{10}|\uparrow_{x}\rangle \langle \uparrow_{x} |+\frac{7}{10}|\downarrow_{z}\rangle \langle \downarrow_{z} |<br />
and with
<br /> s_z=\frac{\hbar}{2}\begin{pmatrix}<br /> 1 & 0 \\<br /> 0 & -1<br /> \end{pmatrix}<br />
Then
<br /> \langle s_z\rangle_{\rho}=\mathbf{Tr}\left[\frac{3}{10}\frac{\hbar}{2}\begin{pmatrix}<br /> 1 & 0 \\<br /> 0 & -1<br /> \end{pmatrix}|\uparrow_{x}\rangle \langle \uparrow_{x} |+\frac{7}{10}\frac{\hbar}{2}\begin{pmatrix}<br /> 1 & 0 \\<br /> 0 & -1<br /> \end{pmatrix}|\downarrow_{z}\rangle \langle \downarrow_{z} |\right]<br />
Now I need help with how to compute the above...
May I have some help?
Thanks
Homework Statement
The most general observable is a density matrix. Generally it is a non-negative self-adjoint operator with trace 1. It has the general form
<br /> \rho=\sum_{n}p_n |n\rangle\langle n|<br />
where p_n is a classical probability distribution (\sum_{n} p_n=1,\; 0\leq p_n \leq 1) and |n\rangle\langle n| are projection operators that are not necessarily orthogonal. \rho represents a classical statistical ensemble of quantum states where the state |n\rangle appears with probability p_n. The ensemble average of an operator O is an ensemble of states described by a density matrix \rho is
<br /> \langle O \rangle_{\rho}=\mathbf{Tr}(O\rho )<br />
Physically this is the average of a number of measurements of O in a classical probability distribution of different states. Consider a polarized beam of protons where 30% of the protons have spin up in the x-direction and 70% have spin down in the z direction. Find the density matrix for this ensemble and compute the ensemble average of s_z in this ensemble of protons.
Homework Equations
<br /> \mathbb{I}=\sum_{n}|n\rangle\langle n|<br />
The Attempt at a Solution
I set up the density matrix like this
<br /> \rho=\frac{3}{10}|\uparrow_{x}\rangle \langle \uparrow_{x} |+\frac{7}{10}|\downarrow_{z}\rangle \langle \downarrow_{z} |<br />
and with
<br /> s_z=\frac{\hbar}{2}\begin{pmatrix}<br /> 1 & 0 \\<br /> 0 & -1<br /> \end{pmatrix}<br />
Then
<br /> \langle s_z\rangle_{\rho}=\mathbf{Tr}\left[\frac{3}{10}\frac{\hbar}{2}\begin{pmatrix}<br /> 1 & 0 \\<br /> 0 & -1<br /> \end{pmatrix}|\uparrow_{x}\rangle \langle \uparrow_{x} |+\frac{7}{10}\frac{\hbar}{2}\begin{pmatrix}<br /> 1 & 0 \\<br /> 0 & -1<br /> \end{pmatrix}|\downarrow_{z}\rangle \langle \downarrow_{z} |\right]<br />
Now I need help with how to compute the above...
May I have some help?
Thanks
